Here, we present a mechanism for tailoring the photonic band structure of a quarter-wave stack without changing its physical periods by embedding conductive sheets. Graphene is utilized and studied as a realistic, two-dimensional conductive sheet. In a graphene-embedded quarter-wave stack, the synergic actions of Bragg scattering and graphene conductance contributions open photonic gaps at the center of the reduced Brillouin zone that are nonexistent in conventional quarter-wave stacks. Such photonic gaps show giant, loss-independent density of optical states at the fixed lower-gap edges, of even-multiple characteristic frequency of the quarter-wave stack. The conductive sheet-induced photonic gaps provide a platform for the enhancement of light-matter interactions.